Finite group schemes of p-rank ≤ 1

Abstract

Let G be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalization of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp(G) of G and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

Varieties of elementary abelian Lie algebras and degrees of modules

“…. The interested reader is referred to [13] for a description of the restricted Lie algebras of p-rank rk p (g) ≤ 1.…”